Related papers: On cyclic DNA codes over $\mathbb{F}_2+u\mathbb{F}…

Let $\mathbb F_q$ be a finite field, where $q$ is an odd prime power. Let $R=\mathbb{F}_q+u\mathbb{F}_q+v\mathbb{F}_q+uv\mathbb F_q$ with $u^2=u,v^2=v,uv=vu$. In this paper, we study the algebraic structure of $(\theta, \Theta)$-cyclic…

In this article, for a finite field $\mathbb{F}_q$ and a natural number $l,$ let $\mathcal{R}$ denote the product ring $\mathbb{F}_q^l.$ Firstly, for an automorphism $\Theta$ of $\mathcal{R},$ a $\Theta$-derivation $\Delta_\Theta$ of…

There is a one-to-one correspondence between $\ell$-quasi-cyclic codes over a finite field $\mathbb F_q$ and linear codes over a ring $R = \mathbb F_q[Y]/(Y^m-1)$. Using this correspondence, we prove that every $\ell$-quasi-cyclic self-dual…

Let $R=\mathbb{Z}_4+u\mathbb{Z}_4,$ where $\mathbb{Z}_4$ denotes the ring of integers modulo $4$ and $u^2=0$. In the present paper, we introduce a new Gray map from $R^n$ to $\mathbb{Z}_{4}^{2n}.$ We study $(1+2u)$-constacyclic codes over…

In this work, extension theorems are generalized to self-dual codes over rings and as applications many new binary self-dual extremal codes are found from self-dual codes over F_2^m+uF_2^m for m = 1, 2. The duality and distance preserving…

In this paper, we study $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-$(1+u)$-additive constacyclic code of arbitrary length. Firstly, we study the algebraic structure of this family of codes and a set of generator polynomials for this family as a…

Let $r,s,t$ be three positive integers and $\mathcal{C}$ be a binary linear code of lenght $r+s+t$. We say that $\mathcal{C}$ is a triple cyclic code of lenght $(r,s,t)$ over $\mathbb{Z}_2$ if the set of coordinates can be partitioned into…

We introduce a Gray map from $\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}$ to $\mathbb{F}_{2}^{2m}$ and study $(1+u)$-constacyclic codes over $\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}},$ where $u^{2}=0.$ It is proved that the image of a…

In this paper, we study cyclic codes over the Galois ring ${\rm GR}({p^2},s)$. The main result is the characterization and enumeration of Hermitian self-dual cyclic codes of length $p^a$ over ${\rm GR}({p^2},s)$. Combining with some known…

In this article, we construct new non-binary quantum codes from skew constacyclic codes over finite commutative non-chain ring $\mathcal{R}= \mathbb{F}_{p^m}[v]/\langle v^3 =v \rangle$ where $p$ is an odd prime and $m \geq 1$. In order to…

Let $F_p$ be the prime field with $p$ elements. We derive the homogeneous weight on the Frobenius matrix ring $M_2(F_p)$ in terms of the generating character. We also give a generalization of the Lee weight on the finite chain ring…

In \cite{anote}, Wu and Shi studied $ l $-Galois LCD codes over finite chain ring $\mathcal{R}=\mathbb{F}_q+u\mathbb{F}_q$, where $u^2=0$ and $ q=p^e$ for some prime $p$ and positive integer $e$. In this work, we extend the results to the…

We first define a new Gray map from $R=\mathbb{Z}_4+u\mathbb{Z}_4$ to $\mathbb{Z}^{2}_{4}$, where $u^2=1$ and study $(1+2u)$-constacyclic codes over $R$. Also of interest are some properties of $(1+2u)$-constacyclic codes over $R$.…

In this paper, we give a matrix construction method for designing DNA codes that come from group matrix rings. We show that with our construction one can obtain reversible $G^k$-codes of length $kn,$ where $k, n \in \mathbb{N},$ over the…

For $\lambda$ an $n$-th power of a unit in a finite chain ring we prove that $\lambda$-constacyclic repeated-root codes over some finite chain rings are equivalent to cyclic codes. This allows us to simplify the structure of some…

Let $\texttt{R}$ be a commutative finite chain ring of invariants $(q,s)$ and $\Gamma(\texttt{R})$ the Teichm\"uller's set of $\texttt{R}.$ In this paper, the trace representation cyclic $\texttt{R}$-linear codes of length $\ell,$ is…

Coterm polynomials are introduced by Oztas et al. [a novel approach for constructing reversible codes and applications to DNA codes over the ring $F_2[u]/(u^{2k}-1)$, Finite Fields and Their Applications 46 (2017).pp. 217-234.], which…

A Z2-triple cyclic code of block length (r,s,t) is a binary code of length r+s+t such that the code is partitioned into three parts of lengthsr,s andt such that each of the three parts is invariant under the cyclic shifts of the…

Binary self-dual codes with large minimum distances, such as the extended Hamming code and the Golay code, are fascinating objects in the coding theory. They are closely related to sporadic simple groups, lattices and invariant theory. A…

Generalizing even-like duadic cyclic codes and Type-II duadic negacyclic codes, we introduce even-like (i.e.,Type-II) and odd-like duadic constacyclic codes, and study their properties and existence. We show that even-like duadic…